In (Adv. Math. 394, 108052, 2022) V. Gélinas introduced a homological invariant, called delooping level (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an Artin algebra. We compare this new tool with other dimensions as the finitistic dimension or the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{\phi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">ϕ</mi> </mrow> </math></EquationSource> </InlineEquation>-dimension (where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{\phi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">ϕ</mi> </mrow> </math></EquationSource> </InlineEquation> is the first Igusa-Todorov function), and we also generalize Theorem 4.3. from Gélinas (Proc. Amer. Math. Soc. 149(12), 5001–5012, 2021) to truncated path algebras (Theorem 4.18). Finally, we show that for a monomial algebra <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </math></EquationSource> </InlineEquation> the difference <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\text{ dell }\varvec{(A)} \varvec{-} \text{ Findim }\varvec{(A)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>dell</mtext> <mspace width="0.333333em" /> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mrow> <mo mathvariant="bold">-</mo> </mrow> <mspace width="0.333333em" /> <mtext>Findim</mtext> <mspace width="0.333333em" /> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> can be arbitrarily large (Example 4.22).

Delooping Levels

期刊论文
发表日期：

2026年2月13日

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Marcos Barrios,
Marcelo Lanzilotta,
Gustavo Mata

摘要

In (Adv. Math. 394, 108052, 2022) V. Gélinas introduced a homological invariant, called delooping level (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an Artin algebra. We compare this new tool with other dimensions as the finitistic dimension or the $\varvec{\phi }$ $ϕ$ -dimension (where $\varvec{\phi }$ $ϕ$ is the first Igusa-Todorov function), and we also generalize Theorem 4.3. from Gélinas (Proc. Amer. Math. Soc. 149(12), 5001–5012, 2021) to truncated path algebras (Theorem 4.18). Finally, we show that for a monomial algebra $\varvec{A}$ $A$ the difference $\text{ dell }\varvec{(A)} \varvec{-} \text{ Findim }\varvec{(A)}$ $dell (A) - Findim (A)$ can be arbitrarily large (Example 4.22).